Hilbert’s Hotel

Infinity. Simply thinking about it is enough to drive people insane – no literally, Georg Cantor the man who more or less wrote the book on infinity was regularly hospitalised for his mental health.

Though that may have been more to due with the criticism he received for working with the infinite. Leopold Kronecker, another mathematician of the time, described Cantor as a “scientific charlatan”, a “renegade” and a “corruptor of youth”. There were many mathematicians and philosophers who were equally appalled by Cantor’s work but there did stand some who supported his theories. David Hilbert famously stated “No one shall expel us from the Paradise that Cantor has created.”

It is with that view that he developed a series of problems based around a hotel with an infinite number of rooms.

Here are the constraints:

You must know what room every guest is staying in

No rooms can be shared

No rooms can be built

No guests are ever turned away

The first problem says that the hotel is completely occupied but another guest arrives. How can you accommodate this final guest?

The answer is simpler than you might think.

Move all guests along one room and put the new guest into room 1. All constraints are fulfilled, no rooms have been shared or built, the new guest is accommodated and if you wanted to know where the guest who was in room 343 is now, you could with confidence say room 344.

But wait, did we not just show that ∞ + 1 = ∞? The hotel was already infinitely full and we added one more so this statement must be true.

That is by far not the weirdest thing found in these problems.

Take problem 2, the hotel is again full but this time a special infinite coach arrives with seats numbered 1, 2, 3 etc. How can these be accommodated?

Again a surprisingly simple answer can arise.

Move every individual to 2 times there room number and put the coach load into the gaps.

So ∞ + ∞ = ∞

Lets try an infinite coach with seats evenly numbered 2, 4, 6 etc.

Just half the numbers and put them in that room.

As evens represent half of all integers this also suggests ∞/2 = ∞

In fact there are many different ways of making that particular size of infinity called ‘countable infinites’.

There is a larger infinities called ‘Uncountable infinities’ and then infinitely more infinitely larger than that.

Well then. I hope that sufficiently confused everyone involved. I’m now off to go rock back and forth in a corner whilst drawing the symbol ∞ over and over again until I fall asleep.

P.S. Hi, this has been my first post. Thank you very much if you have read down this far. I study Maths, Further Maths, Physics and Mandarin at Sixth Form and I intend to study maths at Uni.